a non-linear approach to configuration interaction

Volume 133, number 2 CHEMICAL PHYSICS LETTERS 9 January 1987

A NON-LINEAR APPROACH TO CONFIGURATION INTERACTION. THE LOW-RANK CI METHOD (LR CI)

Jeppe OLSEN, Per-Ake MALMQVIST, Bjijrn 0. ROOS, Roland LINDH and Per-Olof WIDMARK Department of Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden

Received 10 November 1986

An alternative to the linear CI expansion is presented, which is based on the spectral resolution of the double excitation CI coeffkients arranged as a symmetric matrix. The resulting energy expression is a quartic function of the new variables, where the double excitation operator is given as fz = Xyw,( &)2 , where 2, is a single excitation operator: 2, = I&d:. I&,. Since M is a small number, the number of variables is reduced considerably compared to normal SDCI, with only little loss in accuracy. A program for non-linear CI calculations has been written and is presented with results for Hz0 and N2. The method can easily be extended to include orbital optimization and cluster terms in the wavefunction, still yielding a fully variational approximation to the Schro- dinger equation.

1. Introduction

The configuration interaction (CI) method is today a well established procedure for obtaining correlated wavefunctions [ 11. It is used to solve many types of electronic structure problems in different areas of chemistry, often in conjunction with multiconfigurational SCF (MC SCF) methods. The introduction of direct methods fifteen years ago [ 2,3 ] made it possible to treat large CI expansions, comprising of the order of 1 O6 terms. An illustrative example is the recent full CI calculation on the Ne atom, where the wavefunction is expanded in 2360757 configuration state functions (CSFs), using a ( 5s, 3p, 2d) CGTO basis set [ 41. The energy obtained accounts for about 75Oh of the valence correlation energy.

The direct CI method was originally limited to a closed-shell reference configuration [ 21. The introduction of the unitary group approach by Shavitt [ 51 made it possible to extend the direct methods to a wavefunction comprising single and double replacements from a set of reference configurations with arbitrary spin multiplic- ity [ 61. It was this development that turned the direct configuration interaction methods into a general tool for a wide range of chemical applications.

The variational CI method has, however, two serious drawbacks. One is the slow convergence of the linear expansion method. The second is the size-consistency problem [ 71. This latter problem can, at least partly, be overcome by using a multiconfigurational reference space. The cluster corrections can also be approximately accounted for by the coupled electron pair approximation (CEPA) [ 81, or by the recently introduced coupled pair functional (CPF) method [ 91, or by using the simple Davidson correction to the energy [ lo]. The full CI calculation of Bauschlicher et al. [ 41 shows, however, that these methods will only approximately account for the cluster corrections. Further, no strict extension of these methods to the case of a multiconfigurational reference space exists today.

A good illustration of the size bottleneck in the CI method is given by the Cr2 molecule. It has been estimated [ 111 that about 57 million CSFs would be necessary to build a CI wavefunction which would account for the bond properties of this molecule. Recently it has been suggested that empirical corrections of the two-electron integrals could be used to correct for the large intra-atomic correlation effects on the bonding, as an alternative

0 009-26 14/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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to large-scale CI calculations [ 121. Even if such an approach could work in specific cases, it is not in line with the natural development of ab initio methods in quantum chemistry.

An alternative to the linear CI expansion method is suggested in the present contribution, where the information of the correlated wavefunction is concentrated into a much smaller set of variables. The method is based on a spectral resolution of a symmetric matrix formed from the double-replacement Cl coefficients. The eigenvectors of this matrix are the new variables, and the basic assumption behind the method is that the information inherent in the matrix can be concentrated into a few of these eigenvectors, corresponding to eigenvalues not close to zero. The method has been partly inspired by a suggestion by Dacre and Watts [ 131 to include a double- excitation MC SCF wavefunction with coefftcients constructed as products of a new set of single-excitation coefficients. For Be and HZ0 85% of the possible correlation energy was obtained with such a wavefunction. The method was, however not pushed further, probably due to convergence problems in the non-linear optimization procedure.

The resulting energy expression is a quartic function of the new variables - or up to the power 2n if excitation levels up to the order n is included in the CI expansion. Non-linear optimization methods must therefore be used to optimize the variables. The recent developments within the MC SCF method have, however, resulted in efficient second-order procedures based on the Newton-Raphson method, which can also be applied here (see for example the review [ 141). The basic method will in this article be illustrated for the case of a closed- shell reference state plus all single- and double-replacement CSFs. It is straightforward to extend the method to include variationally cluster corrections up to arbitrary order. Parameters for orbital optimization can also easily be included. Ultimately the method suggested thus results in an MC SCF procedure with a wavefunction of the form

I~)=exp(~,++~)IO), (1)

where F, and TZ are the single- and double-excitation operators and IO) is the single reference ground state. In a forthcoming paper it will be shown that a corresponding scheme can also be developed, where IO) is a multiconfigurational wavefunction obtained in a preceding MC SCF calculation with a limited active orbital space, for example a CAS SCF wavefunction [ 15 1.

2. The spectral resolution of the CI matrix

The wavefunction under consideration in this section is of the form

I~>=(1+~~++2)10),

where the single-excitation operator F, has the form

f1 = CSzAai 3 W

(2)

(3)

where E,, are the orbital excitation operators. Orbitals occupied in the reference configuration (internal orbitals) are labelled i, j, k, I..., while the external orbitals are labelled a, b, c, d, . . . . The double excitation operator Z$* is similarly given as

pZ = ,~bcicqbEazgbj * (4)

No restriction is imposed on the summation index. Thus the same excitation occurs twice with the coefficients &,b and C,bra, respectively. The wavefunction depends on the sum only of these two coefficients. They can therefore be assumed to be equal:

ciqb = cjbm * (5)

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Note that the two intermediate spin couplings are not separated in pZ;. This can formally be achieved by introducing the coefficients

C&b = cm,, f ctb,a , (6)

where the two signs correspond to singlet ( + ) and triplet ( - ) coupling of the external electron pair, respectively. Such a separation, however, leads to a more complicated form of pZ and will not be used here.

The coefficients Cia,b can be considered as a symmetric matrix C with row index ia and column indexjb. This matrix can be diagonalized. The coefficients (5) then take the form

ctajb = Cwda&b 3 (7) I

where d’ are the eigenvectors with eigenvalues oI. FZ can thus formally be written in the form

&~“,@“. (8)

This is the basic equation of the method suggested in the present work. 2’ is a single-excitation operator similar to F,;:

(9)

Eqs. (8) and (4) are obviously equivalent if the sum in (8) is allowed to run over all eigenvalues of the matrix C. It will, however, be demonstrated below that most of the eigenvalues oI are close to zero, that is, only a few terms are needed in (8) in order to reproduce FZ to high accuracy. The wavefunction (2) can then be written in the form

IY)= I+s^+ $fB1(&)2 IO\)) ( I=1 > (10)

where M is a small number. Alternatively the exponential ansatz (1) can be used, as is demonstrated below. The variational parameters in (10) are s and d’, which will be treated as matrices with elements Sia and d:=. The eigenvalues mr can be incorporated into d’ if the sign is known, by relaxing the normalization condition. By far the most important contributions to (8) are obtained from the eigenvectors corresponding to negative eigenvalues or, as can easily be demonstrated using second-order perturbation theory.

The number of elements in one vector dz is n,n,, where n, is the number of internal orbitals and n, the number of external orbitals. The total number of variational parameters determining the doubly excited part of the wavefunction is approximately equal to N1 =Mn,ne (it is actually slightly smaller because of the orthogonality constraints), which can be compared to the total number of independent CI coefficients, N2 = 1 nin,( nine + 1) . Thus the size of the problem has been reduced to the square root of the original size, provided of course that it4 is a reasonably small number. The reduction in size will be orders of magnitude larger when higher than double excitations are included into the wavefunction.

The excitation operators & in (10) can be used to rotate the internal orbitals. The double excitation wavefunction can thus be considered as an expression in a set of non-orthogonal virtual orbitals &,,

(11)

In this sense (10) can be regarded as a generalization of Meyer’s PNO CI method [ 161 with the difference that the present method is completely variational and can easily be extended to include contributions from clusters of higher order: it can be made size consistent.

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3. The energy expression

The correlation energy is obtained from the variational energy expression:

EXWr=<~l~-~oIWW~), (12)

where E,, is the reference energy. Using (10) for the wavefunction Y, EC,, becomes a quartic function of the parameters &. In order to compute optimal values for all parameters in (lo), and eventually also orbital rotation parameters (MC SCF), the energy and its first and second derivatives with respect to s and d’ has to be evaluated. Once the energy expression in known it is a trivial task to evaluate the expressions for the gradient and the Hessian. The basic formulas needed to evaluate the energy will be given in this section. Ultimately also multiple excitations will be included in the wavefunction. Therefore the following general matrix element will be evaluated:

Hm,=(OJti”fi-Eoi”~O) (m<n), (13)

where

fi= CUiaJ??*a p t^= Cl?lalQai . (14) *,a W

Using the commutation relations for the generators &,, (13) can be rewritten in the form

where

A,=[fi,A], fi,=[a,[a,Ei]].

The excitation operators S,, are defined as

&]o)=aV]o) .

They can be computed recursively according to the formula

21Tr(w) &_l,~_I + (;) ( k;l ) icL,_z

+(k-I)

(15)

(16)

(17)

(18)

where the following operators have been defined:

?=[[ti,f] ,i] ) a’=[& p.i,i]]) &[a,?] U9a)

and the matrix

w=tu+. (19b)

Only the operators Sk1 for k= I, I- 1, and I- 2 are needed. S), is represented by a scalar S,, which is the overlap integral

&(O(~‘t^l(O) . (20)

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For k= l- 1, Sk, is a single-excitation operator

i,a

and for k= I- 2 a double-excitation operator is obtained which can be represented as

(21)

(22)

Explicit expression for the representations of the operators $, are given in the appendix. Using (15 ) and (18) the matrix elements H,, can now be evaluated for the three non-zero cases m= n, n - 1, and n - 2. The result can be expressed in terms of traces of matrix products involving the matrices t, u, the Fock matrix F and a matrix V( t”) defined from the matrix elements

~(tn)w=Ct,:~[2(P41ia)-(paIiq)l f i,a

(23)

The two-electron integral combination within the brackets in (23) is an element of the super-matrix used to construct the Fock operator in closed-shell Hartree-Fock theory. The matrix V can therefore be computed directly from a list of atomic two-electron integrals, or simpler from a list of super-matrix elements. Note, however, that the list must contain all symmetry blocks, not only those used in calculating the Fock matrix. This is an extremely important result. Its implication is that the present CZ method is an n4 process (n is the number of A0 basis functions), and no two-electron integral transformation is needed.

The following expressions are obtained for the three types of non-zero matrix elements:

Z&-D, =2a2, Tr[tIWt(t%))l , (24)

H,,_,,n =2(n- I),$, {Tr[utt$~)Vt(ti~))l + Tr[uttWt(t~~)l

(25)

H,,=2n(n- 1) 5 {Tr[utt{$Vt(utt&))] +Tr[utj:)Vt(ut$Z)t)] -Tr[utl~)‘V+(u+t~~))l a=1

(26)

In these formulas it is implicitly understood that matrix products and traces are taken over the appropriate index range for the different matrices (e.g. tut is a matrix with both indices running over the internal orbitals; the Fock matrix in the last term of (26) contains elements with both indices internal, etc.). The calculation of the energy (and its derivatives) can be performed efficiently on a vector processor, since all expressions contain only traces of products of matrices. The expressions for the derivatives of the energy can easily be evaluated from the formulas (24)-(26). The explicit formulas for the gradient and the Hessian will therefore not be presented.

Using the general expressions given above it is straightforward to evaluate the correlation energy, which is given as

E corr =(OI~+(Z%-Eo)~+~O)/(OI~+f"IO), (27)

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Volume 133, number 2 CHEMICAL. PHYSICS LETTERS 9 January 1987

where F is given by (10). The extension to include cluster corrections to the energy is obvious: replace p by @‘F= exp( fi. The corresponding correlation energy can be evaluated to arbitrary order in T by means of the same general formulas (24)-( 26). Truncation of fat fourth order is already in most cases a very good approximation to the infinite expansion:

(28)

The corresponding energy is determined variationally by means of (27). The procedure is not completely size consistent, when a truncated form of exp( ?=) is used. The error is, however, expected to be small and can be decreased by including more terms in (28). The great advantage of using a variational procedure is twofold: the energy is an upper bound, but more important is the fact that 6E/6p, = 0 (pi being the variational parameters of the system). The simplifications this leads to in calculations of molecular properties are well known. It is especially important for calculations of molecular gradients and Hessians.

4. Programming considerations

A Fortran 77 computer code for the low-rank CI (LR CI) calculations has been written and tested on the FPS- 164 computer. The preliminary version, for which some test results will be presented below, performs an optimization of a low-rank SDCI wavefimction with a closed-shell reference configuration. The computational procedure can be outlined as follows:

Step 0. Compute one-electron integrals and a two-electron supermatrix P with elements

P ,,~=@'iK~)--(tiiK~). (29)

Note that this matrix has to be evaluated for all symmetry combinations of the basis functions, and that it cannot be symmetrized, as is done in Hartree-Fock theory:

P ~YKAZPJWAK * (30)

The supermatrix is stored in the form of arrays in the second index pair, facilitating the formation of the V matrices as simple dot products. This preliminary step also includes a determination of molecular orbitals, usually from an SCF calculation, and a guess of starting values for the d’ matrices. While zero can be used as starting values for the elements of the s matrix, this is not possible for d’, since it corresponds to a stationary point on the energy surface. One possible way is to obtain the starting d’ as the M lowest eigenvectors of the MO supermatrix, with elements P,,$, which corresponds to the Hessian of the system for d’= 0, and compute the wI by solving a small eigenvalue problem.

This procedure was tried but was found not to give very good starting values for the d’. An ideal procedure would be to obtain starting parameters by first-order perturbation theory, that is, by diagonalizing the matrix P&( E, - c, + $, - 6,)) where E are orbital energies. Such a calculation cannot, however, be performed without carrying through a two-electron integral transformation. Starting values for d’ were instead obtained by diagonalizing the slightly modified matrix Pa&j/ [ ( en- cl)( cb- c,)] “2. The calculation can then be performed in A0 basis by scaling the d’ and the corresponding eigenvalues. The d’ values obtained as eigenvectors of the above matrix were used in a subsequent step which determined the single excitation coefficients s,, and the wl, using the fact that the total energy is a quadratic function of these parameters. Using these starting parameters the final LR SDCI calculation converged in lo-30 iterations for the cases discussed in the next section.

Step I. Compute the energy and the energy gradient. The time-consuming step is the construction of the V matrices (23). The number of such matrices is of the order MZ (cf. eqs. (24)-(26) with ut=dr and t=dJ). The total computational effort is therefore proportional to M2n4, where n is the number of A0 basis functions. In a normal direct CI calculation the number of operations is proportional to n? n," , which may be a smaller number, since Mprobably has to be larger than the number of internal orbitals n,. On the other hand no integral

96


transformation is needed since V can be constructed directly from the A0 supermatrix file. Double buffering and asynchronous reading of this file makes the calculation of the V matrices a completely CPU bound procedure.

Step 2. Solve the Newton-Raphson equations, using the techniques developed in modern MC SCF theories [ 141. Obviously the Hessian is never explicitly constructed, only a set of update vectors G6. The construction of these update vectors is essentially equivalent to the calculation of the energy gradient, and involves as the time-consuming step again the construction of a number of V matrices. In the present preliminary version of the program a more primitive method has been implemented involving a linear search step and the construction of the Hessian through an update procedure using subsequently computed gradients. Thus the explicit calculation of the Hessian is avoided.

Steps 1 and 2 are repeated until convergence, and the calculation is finished by computing natural orbitals and occupation numbers. An MC SCF step can easily be included by introducing a new matrix x such that exp(x) describes the rotation of internal and external orbitals into each other. The extra computational effort is small, but the result is a wavefunction where all parameters are variationally determined. It should be empha- sized that such an orbital optimization is more important there than for a full-rank SDCI wavefunction, since it will partly correct for the missing part of the correlation energy due to the truncation of the eigenvector space for the CI matrix ( 7).

5. Some test calculations

The results of a series of test calculations are discussed in this section. The main purpose of these studies has been to study the convergence of the energy and wavefunction towards the full SDCI result when the number of d vectors in the expansion (10) is increased. The main emphasis will be on energy, but for Nz some other properties are also studied. The different calculations are labelled by the number of d vectors employed in each irreducible representation of the point group used, which was DZh for Ne and Nz, and CZV for H20. Thus a label (4,2, 1, 1) in a water calculation means that eight d vectors are used: four in symmetry a,, two in symmetry bz, one in symmetry bl, and one in symmetry a2. A normal SDCI calculation will be labelled FR SDCI (full-rank SDCI). The results are presented as computed correlation energy, explicitly and in percent of the FR SDCI correlation energy. A measure of the size of the LR SDCI parameter space is obtained as the quotient between the number of variables and the number of independent coefficients in a normal double-excitation CI calculation. The quotient is given in the tables in percent.

The first set of calculations were performed on the neon atom. Here the purpose was to study the effect of increasing the A0 basis set. The basis sets used in the full CI calculations of Bauschlicher et al. [ 41 were used, in order to compare the present results with their SDCI energies. Only one d vector in each symmetry was used in these calculations. The results are presented in table 1. They show how the energy varies with the basis set. A

Table 1 LR SDCI results for the neon atom with one d vector in each symmetry

Basis set a) LR SDCI energy (au)

FR SDCI a) energy (au)

% % of the parameter space b,

4~2~ 0.098293 0.099530 98.7 47.6 5S,3P 0.125666 0.134751 93.3 30.0 6s,4~ 0.128396 0.138889 92.4 21.8

4s,2p,ld 0.162415 0.174133 93.3 29.4 5s,3p,ld 0.185674 0.210989 89.4 22.2 6s,4p,ld 0.191896 0.216032 88.8 16.5 5s,3p,2d 0.200617 0.235733 85.1 17.0

‘) From ref. [ 41. ‘) The number of LR SDCI parameters divided by the number of FR SDCI parameters (doubles only).

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larger basis set gives a smaller relative parameter space compared to FR SDCI (for a given number of d vectors), and also a decreasing relative correlation energy. The smallest DZ basis set gives 98.7% of the FR SDCI result, but the parameter space used is also 47.6% of the full SDCI space. This latter number decreases rapidly as the basis set increases, while the correlation energy decreases towards a constant value. For neon a correlation energy which is about 8OW of the FR SDCI energy would be obtained with one set of d vectors, as the basis set is increased towards infinity.

Table 2 presents the results from a set of calculations performed on the water molecule at experimental geom- etry. An extended basis set was used, consisting of seven s-type, four p-type, and two d-type functions for oxygen, and four s-type and two p-type functions for hydrogen. Excitations from the oxygen 1s orbital were not included. In these calculations the number of d vectors were varied in the following way: The first calculation used the set (1, 1, 1, 1). The energy contribution from each d vector can be estimated using a partitioning of the correlation energy in the form

E= (OlAFz 10) .

With Fz from (8) a partitioning oft is obtained as

(31)

M

E=C Cl, I=1

(32)

where eI is the energy contribution from one specific d vector, d’. Eq. (32) is used to decide whether a new calculation shall be performed using one vector more in a given symmetry. The d vector space is increased with one in a given symmetry if the smallest energy contribution in (32) is less than a fixed threshold. For the water calculation presented in table 2 the threshold chosen was - 0.001 au. The calculations then terminate with the

Table 2 LR SDCI results for Hz0 with an extended basis set ‘I, varying the number of dvectors

d vectors per symmetry LR SDCI %of Parameter space energy b, FR SDCI in%of

aI b2 bi a2 FR SDCI

1 1 1 1 -0.161723 67.2 3.9 2 2 2 2 -0.200669 83.4 7.8 3 3 3 2 -0.214812 89.3 11.0 4 4 3 2 -0.221774 92.2 13.3 5 4 3 2 -0.225003 93.6 14.6 6 4 3 -0.227341 94.5 15.9

64 52 40 - 0.240494 100.0 100.0

a) Basis set: (O/7,4,2), (H/4,2). b, FR SDCI calculation.

Table 3 Energy contributions e, for the different d vectors in the (6,4,3,2) calculation on water (table 2)

Symmetry Vector Total energy contribution

1 2 3 4 5 6

aI -0.113979 - 0.024904 -0.011575 -0.005058 -0.002985 -0.000503 -0.159004 b, -0.029422 -0.008872 -0.002534 0.000349 -0.040478 b, -0.023947 -0.003149 0.000339 -0.026757 a2 -0.002119 0.001017 -0.001102

summed contributions: energy - 0.22734 1

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d vector set (6,4, 3, 2), which gives 94.5% of the FR SDCI correlation energy. Obviously this procedure is not recommended at production stage, but is only used to test the convergence of the LR SDCI method. The energy contributions eI obtained in the largest water calculation (6,4, 3,2) are presented in table 3.

By far the most important contribution is obtained for the first vector in symmetry a,. This vector alone accounts for about 50% of the final result. Note that the last vectors in symmetries bZ, b,, and a2 yield positive energy contributions. This is an artefact of the way the energy is divided according to (32). Obviously the addition of these vectors to the variational space will lower the total energy.

The results in table 2 shows that the convergence in the correlation energy is rapid initially, but slows down considerably at the end. The LR SDCI method will have difficulties in attempting to recover the last few percent of the correlation energy. Convergence is, however, going to improve substantially when orbital optimization parameters are added to the variational space.

Similar results are obtained for Nz (table 4). Here a CGTO basis consisting of five s-type, three p-type, and two d-type functions was used. The calculations were performed with the same threshold as used for HzO. The smallest calculations (one d vector in each symmetry) yields 83.5% of the SDCI valence correlation energy using 6.1% of the full variational space. It is clear that the LR SDCI approach becomes more advantageous when the dimension of the system increases, either by extending the basis set or increasing the number of electrons. The Nz calculations were carried out at three internuclear distances: 2.02, 2.07, and 2.12 au. Table 5 gives the bond distances and force constants obtained by a quadratic fti through the three points. A rapid convergence towards the FR SDCI results is obtained for these two properties.

Already the first set of d vectors decreases the error in r,, from the SCF value of 0.06 au to less than 0.01 au. The same rapid convergence is obtained for the force constant. These results indicate that the LR SDCI method should be very efficient for calculations of energy surfaces. A more general conclusion regarding the convergence of molecular properties must await more test calculations. However, some indication can be obtained

Table 4 LR SDCI results for Nz at r,.,,=2.07 au ‘)

Number of d vectors per symmetry (DZh) LR SDCI %of Parameter space energy FR SDCI in % of FR SDCI

1 2 3 4 5 6 7 8

A 1 1 1 1 1 1 1 1 -0.249041 83.5 6.1 B 2 2 2 1 2 2 2 1 -0.216582 92.1 11.2 C 3 2 2 1 3 2 2 1 -0.282745 94.8 13.5 D 3 2 2 1 4 2 2 -0.286611 96.0 14.6 E 42 26 26 14 45 28 28 -0.298409 100.0 100.0

‘) Basis set (N/15,3,2). b, FR SDCI calculation.

Table 5 Bond distance and force constant for N2 obtained from the LR SDCI calculations of table 4 ‘)

Calculation

SCF A B C D E

rm (au) ‘)

2.0356(0.0561) 2.0835(0.0082) 2.0879(0.0038) 2.0896(0.0021) 2.0908(0.0009) 2.0917 -

Force constant (au) ‘)

1.739(0.009) 1.728( -0.002) 1.729(-0.001) 1.730(0.000) 1.730(0.000) 1.730 -

a) Obtained through a quadratic tit to energies for r=2.02,2.07, and 2.12 au, respectively. b, Difference to the FR SDCI result within parentheses.

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from the convergence of the natural orbital occupation numbers. The results are not presented in detail here, but for N2 (at r,, = 2.07 au) the deviation from the FR SDCI values is less than 0.0 1 with only one set of d vectors. The largest calculation (D in table 4) gives deviations which are smaller than 0.001 electron.

The calculations on N2 also serve as a good illustration of the severe limitations inherent in the method suggested by Dacre and Watts [ 131. They proposed to use only one, fully symmetric, d vector. However, in N2 the most important d vector has symmetry ou (symmetry 5 in table 4). A calculation with only one d vector of symmetry type o, (1 ), yields a correlation energy of - 0.032 117 au (at r m=2.07 au) which is only 10.8% of the FR SDCI result.

6. Conclusions

A new approach to configuration interaction calculations for molecular systems has been presented. The major achievement is the concentration of the information inherent in the CI coefficients into a considerably smaller set of variables. Thus it should be possible to treat considerably larger systems with this method than is possible using the conventional CI approach. Even more important is the possibility to include cluster corrections to the wavefunction and the energy from a variational calculation, when the energy is still an upper bound to the exact eigenvalue of the Hamiltonian. An MC SCF step can easily be added which then makes the wavefunction completely variational in all parameters. The importance of this for calculations of energy surfaces, gradients and molecular Hessians is well known. The possibility to use the LR SDCI approach with a multiconfigurational reference space is presently under investigation. When this problem has been solved a very powerful method will be available for studies of electron correlation in molecular systems.

Acknowledgement

The present study has been supported by a grant from the Swedish Natural Science Research (NFR).

Appendix

The explicit expressions for the representations of the operators &, (eq. (9)) will be given here. &,, is represented by a scalar S,, (&, I 0) = S,, IO) ). For n = O-4 we obtain:

Soo=l, (AlI

S,, =2 Tr(w) ,

S,, =8 Tr(w)2 -4 Tr(w2) ,

(A21

(A31

(A4)

&+=384Tr(~)~ Tr(w)-1152Tr(w)’ Tr(w2)

+288 Tr(w2)2 +768 Tr(w) Tr(w3) -288 Tr(w4) ,

where w=tu+.

(A5)

A closed expression can be found for arbitrary values of n: Find all unique partitionings of n as a sum of positive integers, including the case of the single number n, as for example if n = 3: 3, (2 + 1)) and (1 + 1 + 1) . Use the notation of the theory of permutation groups to write each partitioning symbolically as

100


(nl)N’(n2)N* (n/JNk . . . 3

wheren,>nz>...>nk and

(‘46)

N,nr +N*n*+...+N#k=n.

S,,,, (the overlap integral ( 0 I @ii2 I 0) ) is then generally given as a sum over partitionings P,

(A7)

The single-excitation operators $_ l,n are represented by the matrices Sn_-L,n, which for the four smallest values of n are given by

So,1 =t 9 (A9)

Sl,2=4 Tr(w)t-2wt, (A101

S2,3 = 12w2t-24 Tr(w) wt+24 Tr(w)’ t- 12 Tr(w2) t , (All)

S3,4 =96 Tr(w3) t-288 Tr(w2) Tr(w) t-288 Tr(w)2 wt (Al2)

+ 144 Tr(w2) wt+288 Tr(w) w%+ 192 Tr(w)3 t- 144w3t. 6412)

The representation of the double-excitation operator &+, is given by the matrices t{Z) and t&Z’ ((Y = 1, N,). The following explicit expressions are obtained:

n=2: N2 = 1 ) t,, =t21 =t ) (Al3)

n=3: N,=l , tl, =6[Tr(w)t-wt] , tzl=t, (‘414)

n=4: N,=2, t,,=48 Tr(w)2 t-24 Tr(w2) t-96 Tr(w) wt+48w*, (‘415)

tz,=t, t,2=24wt, &=wt.

References

[ 1 ] I. Shavitt, in: Methods of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977). [ 21 B.O. Roos, Chem. Phys. Letters 15 (1972) 153. [ 31 B.O. Roos and P.E.M. Siegbahn, in: Methods of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977). [4] C.W. Bauscblicher Jr., S.R. Langhoff, P.R. Taylor and H. Partridge, Chem. Phys. Letters 126 (1985) 436. [5] I. Shavitt, Intern. J. Quantum Chem. Symp. 11 (1977) 133; 12 (1978) 5. [6] P.E.M. Siegbahn, J. Chem. Phys. 72 (1980) 1467. [7]R.K.Nesbet,Phys.Rev. 109(1958) 1632;Advan.Chem.Phys.9 (1965) 311; 14 (1969) 1. (81 W. Meyer, J. Chem. Phys. 58 (1973) 1017. [9] R. Ahlrichs, P. Scharfand C. Ehrhardt, J. Chem. Phys. 82 (1985) 890.

[lo] E.R. Davidson, in: The world of quantum chemistry, eds. L.R. Daudel and B. Pullman (Reidel, Dordrecht, 1974). [ 111 S.P. Walch, C.W. Bauschlicher Jr., B.O. Roos and C.H. Nelin, Chem. Phys. Letters 103 (1983) 175. [ 121 M.M. Goodgame and W.A. Goddard III, Phys. Rev. Letters 54 (1985) 66 1. [ 131 P.D. Dacre and C.J. Watts, Mol. Phys. 32 (1976) 1437. [ 141 J. Olsen, D.L. Yeager and P. Jorgensen, Advan. Chem. Phys. 54 (1983) 1. [ 151 B.O. Roos, P.R. Taylor and P.E.M. Siegbahn, Chem. Phys. 48 (1980) 157. [ 161 W. Meyer, Intern. J. Quantum Chem. Symp. 5 (1971) 341.

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a non-linear approach to configuration interaction

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